FIG. 1 is a perspective view of a comb-drive microelectromechanical system (MEMS) mirror 10, which is a resonant, electrostatically drivable plant. The MEMS mirror 10 includes an electrically conductive stationary plate 12, which defines an opening 14, which includes alternating protrusions, i.e., “fingers”16a, and spaces 18a along a side 20a of the opening, and which includes alternating fingers 16b and spaces 18b along an opposite side 20b of the opening. The MEMS mirror 10 also includes an electrically conductive mirror plate 22, which is disposed in the opening 14 and is electrically insulated from the plate 12, and includes tabs 24a and 24b, which anchor the plate 22 to the stationary plate 12 via torsion arms 26a and 26b. The mirror plate 22 includes a surface 27 that is coated with a reflective material to form a mirror on the surface, includes alternating fingers 28a and spaces 30a, which are respectively aligned with the spaces 18a and the fingers 16a, and includes alternating fingers 28b and spaces 30b, which are respectively aligned with the spaces 18b and the fingers 16b. Consequently, the fingers 16 are interleaved with the protrusions 28 such that the mirror plate 22 can resonate back and forth about the arms 26a and 26b relative to the stationary plate 12. As discussed further below, one can cause the mirror plate 22 to resonate in a controlled manner by periodically generating a non-zero voltage difference between the stationary plate 12 and the mirror plate 22. This voltage difference generates a periodic electrostatic force that periodically attracts the mirror plate 22 to the stationary plate 12, the majority of this attractive force being between the interleaved fingers 16 and 28. Because the electrostatic force is periodic, it may cause the mirror plate 22 to oscillate back and forth relative to the stationary plate 12, such as in resonance. One may use the oscillating or resonating mirror plate 22 to, e.g., scan or capture an image by sweeping one or more modulated beams of light (not shown in FIG. 1).
FIG. 2A is a plot versus time of a square-wave voltage V(t) with which one can drive the MEMS mirror 10 of FIG. 1 to oscillate in resonance.
FIG. 2B is a plot versus time of the angular displacement, i.e., amplitude, ⊖(t) of the mirror plate 22 of FIG. 1 relative to the stationary plate 12 when one drives the MEMS mirror 10 of FIG. 1 with the voltage V(t) of FIG. 2A. The time Tr is the period of the resonating mirror 22 and is equivalent to a cycle of 2π radians, where the resonant radial frequency ωr of the mirror plate 22 is given by the following equation:ωr=2π/Tr=2πfr  (1)where fr=1/Tr=the resonant frequency of the mirror plate 22. The resonant frequency fr is set by parameters of the MEMS mirror 10, including the stiffness of the torsion arms 26a and 26b and the mass and dimensions of the mirror plate 22.
Referring to FIGS. 1-2B, in operation one first starts the mirror plate 22 resonating, such as oscillating in resonance. Because the mirror plate 22 is relatively light, one may start the plate resonating by merely moving the MEMS mirror 10 or an apparatus (not shown) in which the MEMS is disposed, or by using another known starting technique.
Next, when the mirror plate 22 has a maximum amplitude ⊖ in a first direction (shown as positive amplitude +⊖ in FIG. 3) at time Tr/4 (or at any equivalent time (4n+1)Tr/4, where n is any integer and where (4n+1)Tr/4 is equivalent to (4n+1)π/2 radians), one generates a non-zero constant voltage difference V between the stationary plate 12 and the mirror plate. One may generate this voltage difference by grounding the stationary plate 12 and applying the voltage V to the plate mirror 22, by grounding the mirror plate and applying the voltage V to the stationary plate, or by respectively applying non-zero voltages V1 and V2 (not shown in FIGS. 1-2B) to the stationary and mirror plates such that |V1−V2|=V. Furthermore, techniques for sensing the amplitude ⊖(t) of the mirror plate 22 and for generating the voltage V when ⊖(t)=+⊖ are known, and, therefore, are not discussed.
As discussed above, the voltage difference V generates an attractive force between the interleaved fingers 16 and 28, and this force effectively pulls the fingers 28 back toward the fingers 16 until the mirror plate 22 moves into a zero-displacement position at time Tr/2 when the mirror plate is substantially coplanar with the stationary plate 12 (assuming that the fingers 16 and 28 have substantially the same thickness and that neither of the plates is distorted).
Then, in response to the plate 22 being in the zero-displacement position at Tr/4, one generates a voltage difference V(t)=0 between the plates 12 and 22 such that there is no electrostatic attraction force between the plates. This absence of an electrostatic force allows the momentum of the mirror plate 22 to rotate the mirror plate to the maximum amplitude (shown as negative amplitude −⊖ in FIG. 2B) in a second direction that is opposite to the first direction at time 3Tr/4.
Next, in response to the mirror plate 22 having the maximum negative amplitude −⊖ at time 3Tr/4, one generates V(t)=V.
As discussed above, the voltage difference V generates an attractive force between the interleaved fingers 16 and 28, and this force effectively pulls the fingers 28 back toward the fingers 16 until the mirror plate 22 moves back into the zero-displacement at time Tr.
Then, as discussed above, in response to the plate 22 being in the zero-displacement position, one generates a voltage difference V(t)=0 between the plates 12 and 22 such that there is no electrostatic attraction force between the plates. This absence of an electrostatic force allows the momentum of the mirror plate 22 to rotate the mirror plate back to the maximum amplitude +⊖ at time 5Tr/4.
Next, in response to the mirror plate 22 having the maximum positive amplitude +⊖ at time 5Tr/4, one generates V(t)=V to repeat the above-described cycle.
Because when V(t)=V an electrostatic force exists between the plates 12 and 22, one may call the portions Tr/4→Tr/2 and 3Tr/4→Tr of the period Tr the driving portions P of the period Tr.
In contrast, because when V(t)=0 no electrostatic force exists between the plates 12 and 22, one may call the portions 0→Tr/4 and Tr/2→3rT/4 of the period Tr the relaxation portions R of the period Tr.
Still referring to FIGS. 1-2B, it is sometimes desired to drive the mirror plate 22 to a large amplitude, with a simple velocity profile, at a frequency f that is slightly different, for example, ±5% different, than the resonant frequency fr, and/or to better control the amplitude ⊖(t) or for other reasons. Additionally, it may be desirable to avoid a large voltage V during portions of the normal drive periods P where portions of the MEMS scanner are in close proximity to one another, for example to avoid electrical arcing.
Unfortunately, it is often difficult reach large amplitudes, and difficult to impossible to drive the mirror plate 22 to have a simple velocity profile or at frequencies that are more than a few fractions of a percent different than fr using the square-wave voltage V(t).